Data-Generating Process

FCMs as a Measure of Stress

Movement Data

Hunting Events

FCM Data

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation.

<!– Niko: Still not sold by this - Why is the Defecation Event mentioned?

Thomas (Answer): Because the Faecal Sample is the actual datapoint we end up using. And to make clear why and how timediff is associated with this sample (which is in the same location as the defec event) we should show it here. –>

Relevant Hunting Events

To identify relevant Hunting Events respective to a given FCM Sample, we introduce three selection parameters:

• Gut retention time (GRT) target [hours]: Target Delay between Stress Event and Defecation
• Gut retention time (GRT) thresholds [hours]: maximum temporal Distance between respective Deer and Hunting Event, with the Minimum beeing Zero
• Distance threshold [km]: Maximum spatial Distance between respective Deer and potential Hunting Event

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by one of the three introduced proximity criteria:

• the closest in time to GRT = 19 hours (“closest in time”),
• the closest in space (“nearest”), or
• the one with the “highest score”.

\[ S(d, t) = \frac{10^{10}}{d^2} \times \begin{cases}h \cdot \exp\left( -\frac{(t - p)^2}{2 \sigma_r^2} \right), & \text{if } t \leq p, \\h \cdot \exp\left( -\frac{t - p}{\sigma_f} \right), & \text{if } t > p.\end{cases}\text{where}\begin{align*}S(d, t) & \text{ = Score } \\d & \text{ = Distance } \\t & \text{ = Time Difference } \\p & = \text{peak time = 19 } \\\sigma_r & = \text{rise standard deviation = 2 } \\\sigma_f & = \text{fall standard deviation = 2.5 } \\h & = \text{height scaling factor = 1}\end{align*} \]

Plots

Illustration

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

• Time difference [hours]
• Distance [km]
• Sample delay [hours]
• Pregnant
• Defecation day (between 1 and 366)
• Number of other relevant hunting events

The Models

A Generalized Additive Mixed Model

Gamma: Log Link

• Log link for interpretability.

• Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer.

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\sim& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

A Generalized Additive Mixed Model

Gamma: Log Link

Last

A Generalized Additive Mixed Model

Gamma: Log Link

Nearest

A Generalized Additive Mixed Model

Gamma: Log Link

Highest score

B XGBoost

TBD